proposition 3

jump_lin_id/id_paper.tex

@@ -103,7 +103,7 @@
 
 \newcommand{\fourier}[2]{\mathcal{F}_{#1}\big(#2\big)}
 
-% \linespread{0.99}
+\linespread{0.994}
 %\addtolength{\abovedisplayskip}{-0.5mm}
 %\addtolength{\belowdisplayskip}{-0.5mm}
 
@@ -437,11 +437,17 @@ We formalize the above arguments as
     In the limit $\lambda \rightarrow \infty$ the problem reduces to the LTI problem. The nullspace of the regularization Hessian, which is invariant to $\lambda$, does thus not share any directions with the nullspace of $\tilde{\A}\T \tilde{\A}$ which establishes the equivalence of identifiability between the LTI problem and the LTV problems.
 \end{proof}
 \begin{proposition}
-    Optimization problems \labelcref{eq:smooth,eq:pwlinear} with higher order differentiation in the regularization term have unique global minima for $\lambda > 0$ if and only if the corresponding LTI optimization problem has a unique solution and the Jacobian of the generating system is bounded along the trajectory.
+    Optimization problems \labelcref{eq:smooth,eq:pwlinear} with higher order differentiation in the regularization term have unique global minima for $\lambda > 0$ if and only if there exists a vector $v \inspace{n+m}$ such that
+    \begin{equation}
+        C^{xu}_t v = \bmatrixx{x_tx_t\T & x_tu_t\T \\ u_tx_t\T & u_tu_t\T}v = 0 \;\forall t
+    \end{equation}
 \end{proposition}
 \begin{proof}
     Again, the cost function is a sum of two convex terms and for a global minimum to be non-unique, the Hessians of the two terms must have intersecting nullspaces.
-    In the limit $\lambda \rightarrow \infty$ the regularization term reduces to a linear constraint set, allowing only parameter vectors that lie along a line through time. Let $v \neq 0$ be such a vector and let $L$ be an upper bound on the Frobenius norm of the Jacobian. $v \in \operatorname{null}{(\tilde{\A}\T \tilde{\A})}$ implies that the loss is invariant to the pertubation $k+\alpha v$ for an arbitrary $\alpha$. However, $\exists \alpha : \norm{k+\alpha v} > L$ which establishes the equivalence of identifiability between the LTI problem and the LTV problems.
+    In the limit $\lambda \rightarrow \infty$ the regularization term reduces to a linear constraint set, allowing only parameter vectors that lie along a line through time. Let $\tilde{v} \neq 0$ be such a vector, parametrized by $t$ as $\tilde{v} = \bmatrixx{v & 2v & \cdots & Tv}\T \inspace{T\times(n+m)}$ for arbitrary $v\inspace{n+m}$. $\tilde{v} \in \operatorname{null}{(\tilde{\A}\T \tilde{\A})}$ implies that the loss is invariant to the pertubation $\alpha \tilde{v}$ to $\tilde{k}$ for an arbitrary $\alpha \inspace{}$.  $(\tilde{\A}\T \tilde{\A})$ is given by $\operatorname{blkdiag}(\left\{C^{xu}_t \right\}_1^T)$ which means that $\tilde{v} \in \operatorname{null}{(\tilde{\A}\T \tilde{\A})} \Longleftrightarrow  \alpha t C^{xu}_t v = 0 \; \forall (\alpha,t) \Longleftrightarrow v \in \operatorname{null}{(C^{xu}_t)} \;\forall t$.
+
+
+
 \end{proof}
 
 For the LTI problem to be well-posed, the system must be identifiable and the input $u$ must be persistently exciting of sufficient order~\cite{johansson1993system}.
@@ -525,7 +531,7 @@ The input was Gaussian noise of zero mean and unit variance, state transition no
 
 
 \section{Example -- Non-smooth robot arm with stiff contact}
-To illustrate the ability pf the proposed models to represent the non-smooth dynamics along a trajectory of a robot arm, we simulate a two-link robot with discontinuous Coulomb friction. Additionally, we let the robot establish a stiff contact with the environment to illustrate both strengths and weaknesses of the modeling approach.
+To illustrate the ability of the proposed models to represent the non-smooth dynamics along a trajectory of a robot arm, we simulate a two-link robot with discontinuous Coulomb friction. We also let the robot establish a stiff contact with the environment to illustrate both strengths and weaknesses of the modeling approach.
 
 The state of the robot arm consists of two joint coordinates, $q$, and their time derivatives, $\dot q$. \Cref{fig:robot_train} illustrates the state trajectories, control torques and simulations of a model estimated by solving~\labelcref{eq:pwconstant}. The figure clearly illustrates that the model is able to capture the dynamics both during the non-smooth sign change of the velocity, but also during establishment of the stiff contact. The learned dynamics of the contact is however time-dependent, which is illustrated in \Cref{fig:robot_val}, where the model is used on a validation trajectory where a different noise sequence was added to the control torque. Due to the novel input signal, the contact is established at a different time-instant and as a consequence, there is an error transient in the simulated data.
 \begin{figure*}[htp]